We study the propagation of edge magnetoplasmons by time-resolved current
measurements in a sample which allows for selective detection of edge states in
the quantum Hall regime. We observe two decoupled modes of edge and inter-edge
magnetoplasmons at filling factors close to 3. From the analysis of the
propagation velocities of each mode the internal spatial parameters of the edge
structure are derived.Comment: 4 pages, 4 figures, submitte

F. Hohls

G. Sukhodub

O. I. Kirichek

R. J. Haug

S. A. Mikhailov

V. A. Volkov

English

arXiv

The propagation of edge magnetoplasmons (EMP) in a degenerated two-dimensional electron gas was analyzed. Investigations show that an interedge EMP mode attributed to the excitations of boundary between two regions with constant but different charge densities. The splitting of an incident voltage pulse into several EMP modes propagating with different velocities was also observed. The results show that the internal spatial parameters of the edge structure were derived from the analysis of the propagation velocities of each mode

Sukhodub, G.

Hohls, Frank

Haug, Rolf J.

Institutional Repository of Leibniz Universität Hannover

VOLUME 93, NUMBER 19 P H Y S I C A L R E V I E W L E T T E R Sweek ending5 NOVEMBER 2004Observation of an Interedge Magnetoplasmon Mode in a DegenerateTwo-Dimensional Electron GasG. Sukhodub,* F. Hohls, and R. J. HaugInstitut für Festkörperphysik, Universität Hannover, Appelstrasse 2, D-30167 Hannover, Germany(Received 6 July 2004; published 1 November 2004)196801-1We study the propagation of edge magnetoplasmons by time-resolved current measurements in asample which allows for selective detection of edge states in the quantum Hall regime. At filling factorsclose to   3 we observe two decoupled modes of edge excitations, one of which is related to theinnermost compressible strip and is identified as an interedge magnetoplasmon mode. From the analysisof the propagation velocities of each mode the internal spatial parameters of the edge structure arederived.DOI: 10.1103/PhysRevLett.93.196801 PACS numbers: 73.43.Lp, 73.20.MfFIG. 1 (color online). Schematic view of EMP and IEMPmodes propagating with different velocities and decoupled byan incompressible strip with filling factor   2. Compressiblestrips are shown as gray areas, incompressible liquids as whiteareas, and the hatched region represents a depleted semicon-ductor at the mesa edge.Low-frequency collective excitations of the chargedensity in a two-dimensional electron system (2DES) ina magnetic field travel along the edge of the sample andare called edge magnetoplasmons (EMP’s). Since theirfirst observation by Allen et al. [1] EMP’s have beenextensively studied both theoretically [2– 4] and experi-mentally [4–6]. An interedge magnetoplasmon (IEMP)mode [7,8] attributed to the excitations of the boundarybetween two regions with constant but different chargedensities was experimentally observed up to now only ina classical system of surface electrons on helium [9,10].In a degenerate 2DES under quantum Hall effect con-ditions different density profiles separated by incom-pressible strips with constant charge densities appear[11]. The influence of edge channels (EC’s) on the propa-gation of EMP’s was addressed in experiments onscreened 2DES’s in the frequency domain [5,12] and inthe time domain [6,13]. The splitting of an incidentvoltage pulse into several EMP modes propagating withdifferent velocities was observed in unscreened samples[14]. The additional slower modes were interpreted asacoustic EMP modes [15]. In none of these experimentswith nonclassical systems were IEMP modes identified.We report here the observation of both EMP and IEMPmodes in a screened 2DES in the quantum Hall regime.In our experiment we clearly demonstrate the decompo-sition of the two-stage rise of the observed current pulsein the vicinity of the bulk filling factor   3 into differ-ent modes of edge magnetoplasmons. Using selectivedetection of edge states we can distinguish two wavepackets of edge excitations moving with different veloc-ities near the sample boundary (Fig. 1). The faster onepropagates along the compressible region formed by twostrongly coupled outer compressible strips originatingfrom the lowest Landau level. It gives rise to a fundamen-tal EMP mode. The slower wave packet propagates alongthe inner compressible strip arising from the secondLandau level. This wave packet corresponds to an IEMPmode [7,8].0031-9007=04=93(19)=196801(4)$22.50 A schematic view of the experimental setup is shown inFig. 2(a). An AlGaAs/GaAs heterostructure is patternedinto a T-shaped form with a width of 200 m. The 2DEShas a carrier concentration ns  1:8 1015 m2 and amobility   70 m2=Vs. A long voltage pulse of 800 nsand typically 2.0 mV amplitude with 1 ns rise time isapplied to the source contact S. With the perpendicularmagnetic field pointing into the page it propagates alongthe upper boundary of the sample to the drain contact D1.A Schottky gate G crosses the entire sample and issituated between S and D1. If a negative voltage is appliedto this gate some fraction of the signal is deflected to theother drain contact D2. The distances between source Sand each of the drain contacts D1 and D2 are equal andamount to L  1:56 mm. Thus, the transient currentsdetected at both contacts are directly comparable. Inaddition, a metallic top gate is deposited on top of theentire sample. The distance d between the top gate and the2DES is 105 nm. A thin layer of polymethylmethacrylatecovering the Schottky gate G isolates the gates from eachother in the overlap region. Transmitted pulses are am-plified and recorded by a multichannel digital phosphoroscilloscope (DPO). In such a setup the voltage drop at2004 The American Physical Society 196801-1FIG. 2 (color online). (a) Sketch of experimental setup. Themetallic top gate which covers the whole area of the sample isomitted for clarity. (b) Scheme of selective detection of edgestates. The local filling factor g under the gate G is tuned tog  2. The populated compressible strips are shown as darkareas. (c),(d) Transient currents at B  2:70 T   2:8 mea-sured at contacts D1 and D2 for the entire edge structure (solidlines in both panels), the two outer EC’s (dashed line in (c)),and the inner EC (dashed line in (d)). Sketches along each curveshow the configuration of detected edge states.VOLUME 93, NUMBER 19 P H Y S I C A L R E V I E W L E T T E R Sweek ending5 NOVEMBER 2004the input resistance of the preamplifier is proportional tothe current through the drain contact.The scheme of selective detection of EC’s is depicted inFig. 2(b). A bulk filling factor  is set by the externalmagnetic field. Here we concentrate on   3 since thischoice yields the clearest occurrence of the IEMP. Thecontribution of the entire edge structure can be measuredat both drain contacts D1 and D2. In the first case the gateG remains unbiased (VG  0 mV) and all EC’s are trans-mitted to D1. The local filling factor g under the gate isequal to the bulk value  [upper sketch in Fig. 2(c)]. In thesecond case a negative voltage of VG  350 mV isapplied to the gate which depletes the 2DES underneathcompletely (g  0) deflecting all EC’s to D2 [uppersketch in Fig. 2(d)]. To measure the contribution of eachLandau level by different contacts the filling factor g  2is established under the gate G [this situation is depictedin Fig. 2(b)]. In this configuration the two outer EC’sformed by the lowest Landau level are transmitted toD1 and the innermost EC is deflected to D2.Figure 2 represents the main finding of our study, wherethe decomposition of the two-stage rise of the pulse intotwo single modes is demonstrated. The time-resolvedcurrents for   2:83 detected at contacts D1 and D2are shown in panels 2(c) and 2(d), respectively. The originof the time axis t  0 corresponds to the application ofthe pulse to the source S. The upper solid curves in bothpanels represent the signal carried by the entire edgestructure. Though different contact pairs were used todetect these traces they closely resemble each other. Inboth cases the delay time t0, which is the traveling time ofthe signal between source and drain, is equal to t0 4:63 0:14 ns, and the first steep rise saturates com-pletely after t  30 ns to I1  2:02 0:03e2=hVin,where Vin  2:0 mV is the pulse amplitude. This firststep is followed by a slower rise which develops at t >70 ns and saturates to I  I1 	 I2 with I2  0:920:04e2=hVin for this filling factor.The current associated with edge channels originatingfrom the lowest Landau level only is represented by thelower dashed curve in Fig. 2(c). In its initial part it bears astrong similarity to the upper curve: after t  30 ns thecurrent saturates to 2e2=hVin. For longer times it re-mains constant as in the case of   2. The second slowerrise is completely missing in this trace. The current pulsearrives at D1 as a single wave packet without beingredistributed over the whole edge structure. The initialparts of both curves in Fig. 2(c) reveal minor differences(note the logarithmic scale of t in the figure). Namely, thedashed curve (g  2) is delayed by 1:0 ns and shows aslight change in the slope of the rising part. The first issueis related to the reduced charge density under the gate.This leads to a broadening of the edge channels attributedto the lowest Landau level and as a consequence to a lowerpropagation velocity under the gate. The change in slope196801-2is probably due to the disturbance of the propagation ofedge excitations caused by the negatively biased gate.Nevertheless, the lower curve in Fig. 2(c) reflects themain features of the fast propagating EMP mode.The lower curve in Fig. 2(d) measured at contact D2shows the contribution of the inner EC attributed to theupper Landau level. This slower mode has a delay time oft0  64:0 ns and saturates to I2  e2=hVin after severalhundred nanoseconds. There is no contribution of the fastmode in this measurement, whereas, it resembles verymuch the second slower rise in the solid curve in Fig. 2(d).Here we have detected the pure IEMP mode. To ourknowledge, this is the first observation of an IEMPmode in time-resolved current measurements in a degen-erate 2DES.The observability of decoupled EMP modes dependson the relation between the widths a, l1, and l2 of incom-pressible and compressible strips (see sketch in Fig. 1) andthe distance d from the top gate to 2DES. If the conditiond  a; l1; l2 is fulfilled, the electric field associated with196801-2VOLUME 93, NUMBER 19 P H Y S I C A L R E V I E W L E T T E R Sweek ending5 NOVEMBER 2004each mode can be assumed to be localized within thecorresponding compressible strip and is determined bythe gradient of the charge density across this strip [13].The width of an incompressible strip is substantiallysmaller than the width of the surrounding compressibleones [11]. It is roughly proportional to the gap E in theenergy spectrum of a 2DES in a perpendicular magneticfield. Using the effective g-factor value jgj  0:44 forbulk GaAs and E  gBB we estimate the width a1of the incompressible strip with filling factor 1  1 tobe comparable to the magnetic length. Thus, we do notexpect to observe decoupled transport due to this stripand, indeed, the contribution of the outer two compress-ible strips is represented by a single step in our experiment[lower curve in Fig. 2(c)]. The cyclotron gap E  h!c,being significantly larger, gives rise to a broader incom-pressible strip with a2 > 100 nm facilitating the obser-vation of decoupled edge state transport.Current traces at different magnetic fields around  3 are shown in Fig. 3 for both modes measuredtogether [Fig. 3(a)] and for each mode measured inde-pendently [Fig. 3(b)]. We clearly observe a decrease in thevelocity with magnetic field for both modes, althoughthis is much more pronounced for the slower mode. It isremarkable that the fast mode [Fig. 3(b), D1] does notchange its form considerably in the filling factor range3> > 2 where its amplitude is ruled by 2e2=h. For aquantitative description of the velocities of the fast modewe analyzed the initial parts (t < 70 ns) of the currenttraces corresponding to the entire edge structure[Fig. 3(a)]. The IEMP mode velocities were derivedFIG. 3 (color online). Time-dependent currents measured atB  2:56, 2.64, 2.72, and 2:80 T (  2:99, 2.90, 2.81, and2.73) for the entire edge structure (a) and for the contributionsof the outer and inner compressible regions attributed to EMPand IEMP modes (b). The input pulse voltage is Vin  2:5 mV.196801-3from signals detected at D2 for the case g  2 [lowerset of curves in Fig. 3(b)]. The delay time t0 was obtainedfrom the fit with an exponential decay function [16]. Thevelocities of EMP and IEMP modes calculated from thedelay times by v  L=t0 are presented in Figs. 4(a) and4(b). At  < 3 the velocity of the fast mode decreasesmonotonously achieving a value of 150 km=s at fillingfactor   2. The velocity of the slower mode decreasesproportional to 1=B B0 with B0  2:48 T. For B>2:80 T it cannot be observed any more in our measure-ment window of 800 ns.The velocities of (I)EMP modes can be phenomeno-logically described by v  en=B=0r where n isthe difference in the charge density on both sides of thepropagation region of each mode and   1 is a parame-ter which depends on the shape of the density profile [9].Taking a2 as the separating strip between the two modes,the ratio nEMP=nIEMP grows from 2.0 to 2.73 when themagnetic field increases from 2.55 to 2.80 T. This changecannot explain the experimentally observed larger differ-ence in the velocities: vEMP=vIEMP varies from 3.3 to 17.5in this range of magnetic fields. Therefore, differentshapes of density profiles EMP and IEMP and theirsignificant change with magnetic field have to be takeninto account. Such magnetic field dependence is in accor-dance with the EC picture of the quantum Hall effect,where the compressible regions of varying charge densityare expected to get broader with increasing field [11].To estimate the spatial extension of the density profilefor each mode we use a local capacity approximationdeveloped in Ref. [13]. It connects the group velocity vgof the EMP wave packet with the width l of the propa-gation region and the distance d between the 2DES andthe metallic top gate:FIG. 4. Velocities of edge magnetoplasmons (a) and interedgemagnetoplasmons (b). Right part shows calculated widths ofcompressible regions arising from the first (c) and thesecond (d) Landau levels according to Eq. (1).196801-3P H Y S I C A L R E V I E W L E T T E R S week ending5 NOVEMBER 2004vg @!@ke2hd0rl; d  l; (1)where  is the difference between the filling factors ofthe restricting isolating regions to the left and to the rightof the propagation strip. For the EMP mode   g0  2; for the IEMP we take    g. The magneticfield dependence of the (I)EMP velocities manifests itselfin the increase of the widths l1 and l2 according to Eq. (1).The calculated widths of the propagation regions associ-ated with the EMP and the IEMP modes are shown inFigs. 4(c) and 4(d). The obtained values exceed substan-tially the distance to the top gate for both modes justify-ing the local capacity approach. The width of the outercompressible strip can be estimated from theory [11]. Weobtain l1  190 nm at B  2:5 T increasing to l1 277 nm at B  2:7 T. This agrees quite reasonably withthe experimental result for the EMP mode [Fig. 4(c)]. Theobserved plateaulike structure at  > 3, however, cannotbe appropriately accounted for by a theory, where thewidths of compressible strips depend monotonously onthe magnetic field.In our experiment we observe a distinct transition fromcoupled ( > 3:1; [17]) to decoupled ( < 3:1) transportregimes, when the IEMP mode shows up. The decouplingof the innermost edge state is known to be asymmetricwith respect to the quantum Hall plateau. The authors ofRef. [18] proposed a hybridization of the current-carrying edge states with localized states of the sameLandau level when this level approaches the Fermi energy.Then, the wave function of the innermost EC increases itsextension and reduces at the same time its amplitude nearthe edge. This means, on the one hand, a decrease in IEMPand, on the other hand, an increase in the effective widthof the incompressible strip a which enables the observeddecoupling. In addition, the spin gap governing   3 canbe strongly enhanced by interaction effects influencingthe actual charge distribution.The obtained overall sizes of the edge structurel1 	 l2 are of the same order of magnitude as in previousestimates from, e.g., capacitance measurements [19] andscanned potential microscopy [20]. From our analysis,however, we are able to resolve the internal spatial struc-ture of edge states in a range of magnetic fields wheretheir individual contributions can be unambiguouslyidentified.The decoupled fundamental modes of EMP’s assignedto different Landau levels have been recently theoreti-cally treated in Ref. [3]. It was shown that both modes arenearly undamped and propagate with sufficiently differ-ent group velocities. The chosen confinement potential,however, was sufficiently steep to neglect the formation ofincompressible and compressible strips. Our observedchanges in the velocities which are especially strong forthe IEMP mode suggest that the spatial extent of thecharge density profile connected with each mode mustVOLUME 93, NUMBER 19196801-4also be taken into account. Here, more theoretical work isnecessary.In summary, we employed selective detection of edgestates for time-resolved current measurements under con-ditions of strongly screened Coulomb interactions. Weobserved decoupled EMP and IEMP modes propagatingalong neighboring compressible strips in the vicinity offilling factor 3.The authors wish to thank R. Winkler for a criticalreading of the manuscript and acknowledge financialsupport by DFG and the BMBF.*Electronic address for corresponding author:sukhodub@nano.uni-hannover.de[1] S. J. Allen, Jr., H. L. Störmer, and J. C. M. Hwang, Phys.Rev. B 28, 4875 (1983).[2] V. A. Volkov and S. A. Mikhailov, Sov. Phys. JETP 67,1639 (1988).[3] O. G. Balev and P. Vasilopoulos, Phys. Rev. Lett. 81, 1481(1998); O. G. Balev, P. Vasilopoulos, and N. Studart, J.Phys. Condens. Matter 11, 5143 (1999).[4] Edge Excitations of Low-Dimensional Charged Systems,edited by Oleg Kirichek (Nova Science Publishers,Huntington, NY, 2000).[5] V. I. Talyanskii, A.V. Polisski, D. D. Arnone, M. Pepper,C. G. Smith, D. A. Ritchie, J. E. F. Frost, and G. A. C.Jones, Phys. Rev. B 46, 12427 (1992).[6] N. B. Zhitenev, R. J. Haug, K. v. Klitzing, and K. Eberl,Phys. Rev. Lett. 71, 2292 (1993).[7] S. A. Mikhailov and V. A. Volkov, J. Phys. Condens.Matter 4, 6523 (1992).[8] S. A. Mikhailov, JETP Lett. 61, 418 (1995).[9] P. K. H. Sommerfeld, P. P. Steijaert, P. J. M. Peters, andR.W. van der Heijden, Phys. Rev. Lett. 74, 2559 (1995).[10] O. I. Kirichek and I. B. Berkutov, Low Temp. Phys. 21,305 (1995).[11] D. B. Chklovskii, B. I. Shklovskii, and L. I. Glazman,Phys. Rev. B 46, 4026 (1992).[12] V. I. Talyanskii, D. R. Mace, M Y. Simmons, M. Pepper,A. C. Churchill, J. E. F.Frost, D. A. Ritchie, and G. A. C.Jones, J. Phys. Condens. Matter 7, L435 (1995).[13] N. B. Zhitenev, R. J. Haug, K. v. Klitzing, and K. Eberl,Phys. Rev. B 52, 11277 (1995).[14] G. Ernst, R. J. Haug, J. Kuhl, K. von Klitzing, and K.Eberl, Phys. Rev. Lett. 77, 4245 (1996).[15] I. L. Aleiner and L. I. Glazman, Phys. Rev. Lett. 72, 2935(1994).[16] G. Sukhodub, F. Hohls, and R. J. Haug, Physica(Amsterdam) E 22, 189 (2004).[17] G. Sukhodub, F. Hohls, and R. J. Haug (to be published).[18] B.W. Alphenaar, P. L. McEuen, R. G. Wheeler, and R. N.Sacks, Phys. Rev. Lett. 64, 677 (1990).[19] S. Takaoka, K. Oto, H. Kurimoto, K. Murase, K. Gamo,and S. Nishi, Phys. Rev. Lett. 72, 3080 (1994).[20] K. L. McCormick, M.T. Woodside, M. Huang, M. Wu,P. L. McEuen, C. Duruoz, and J. S. Harris, Phys. Rev. B59, 4654 (1999).196801-4

Observation of an interedge magnetoplasmon mode in a degenerate two-dimensional electron gas

Crossref

Observation of an Interedge Magnetoplasmon Mode in a Degenerate Two-Dimensional Electron Gas

Observation of inter-edge magnetoplasmon mode in a degenerate
  two-dimensional electron gas

Observation of inter-edge magnetoplasmon mode in a degenerate two-dimensional electron gas

Abstract

Similar works

Full text

Available Versions

Institutional Repository of Leibniz Universität Hannover

Crossref